86 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA
For D(ft(x)) is the limit, as h ^ 0 tends to 0, of the mapping
and the result therefore follows from (12.16.5).
The mapping f'to is said to be the derivative at the point t0 of the mapping
tt-+ft (with respect to the topology of simple convergence), or the weak
derivative when F is the field of scalars (and therefore &(E; F) = E').
Remark
(12.16.6.1) Let A be an open set in C and let zi-»/z be a weakly analytic
mapping (12.15) of A into the dual E' of a complex Frechet space F. Then the
same argument as in (12.16.6) shows that there exists a weak derivative zt-+fz
of z\-*fz9 with fz(x) = D(fz(x)) for all xe E, and this weak derivative is
weakly analytic. Moreover, for each a e A and each circuit y contained in
A — {a}, we have the Cauchy formula
(12.16.6.2) ;(* ;?)/«(*)
= J^ {fz(x)dz
2nijy z-a
Conversely, let y be a road in C defined in an interval I = \b, c] in R, and
let z\-+gz be a weakly continuous mapping of y(I) into E'. Then, for z
the mapping
(12.16.6.3)
f 9-
Jy C
is a linear form/z on E, belonging to E'. Indeed, for each x e E the right-hand
side of (12.16.6.3) is the limit of a sequence
n jt=o U - z
where
and
((3.16.5) and (8.7.8)), hence our assertion is a consequence of the Banach-
Steinhaus theorem. By virtue of (9.9.2), fz is weakly analytic in C — y(I), and
for a $ y(I) we can write
(12.16.6.4) fjx) = f CB(X)(Z - a)",
n = 0
the series being convergent in any disk with center a not meeting y(I). The of extension of identities.